Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X₀, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X₀, X) of the subdiagram type whenever X₀ is nonlinear. It remains unsolved whether rigidity holds when (X₀, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X₀, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure \(\overline{\omega} : \mathscr{C}(S) \rightarrow S\) on a uniruled projective manifold \((X,\,{\cal K})\) equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ? emanating from a general point x ∈ S, there exists an immersed neighborhood N_? of ? which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ? X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ? 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X₀ into X which induces an isomorphism on second homology groups. By studying ?*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X₀ ? X. Under the additional hypothesis that all holomorphic sections in Γ(X₀, T_x∣x₀) lift to global holomorphic vector fields on X, we prove that the admissible pair (X₀, X) is rigid. As examples we check that (X₀, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ? ?²ⁿ, n ? 3, and when X₀ ? X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

     

Rational curves of minimal degree and characterizations of projective spaces

In this paper we investigate complex uniruled varieties X whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point x ∈ X form a linear subspace of T_xX. As a first application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wiśniewski's characterizations of . We also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree.     

Inconsistency of GPK + AFA

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X₁ introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X₁ in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. Mathematics Subject Classification: 03E70, 03E10.     

A x -operator on complete riemannian manifolds

In this paper we give a generalisation of Kostant’s Theorem about theA _x -operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operatorA _x =L _x −≡ _x associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA _x -operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA _x does not lie in the holonomy algebra at any point.     

Algebraic morphisms into grassmannians:differential geometry and frobenius

Let Xbe a projective variety (over Spec(K)) and f:X→G(r,v) a morphism to a Grassmannian, i.e. a pair (E,V) where E is a rank r vector bundle on V?H^O(X,E) is a subspace spanning E with dim(V) = v. Here we study the differential properties of f and their relations to a sequence of quotient bundles E→E₁→E₂→of E called the derived bundles of (E,V). In the first 5 sections we study the case X a smooth curve, char(K) >0 (the case char(K) = 0, being due to D. Perkinson). Then we give a general duality theorem for the derived bundles when Xis any normal variety.     

Weak continuity of Riemann integrable functions in Lebesgue-Bochner spaces

In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

Resolution trajectorielle et analyse numerique des equations differentielles stochastiques

Let (ξ_t) be the solution of the S.D.E. (E) of Section 1. Doss [3] has shown the existence of a difFerentiable function h and of a differentiate process parametrized by the process W,γ(W,t), such that: ξ_t = h(γ(W, t), W_t). For all continuous functions u, X_t is defined by: X_t = h(γ(u, t) u_t). We develop a scheme of approximation of X_t (Theorems 2-6 and 3-4). This scheme has th following properties:?

1)it does not explicitly involve γ or h; this property is crucial, because,generally, h is not explicitly known, and its numerical approximation would be costly.

2)it converges to X_t, provided that u has bounded quadratic variation.

3)for u = W, it coincides with a scheme proposed by Milshtein [6] for quadratic-mean approximation.

Further, we give an estimate of the error due to this scheme.     

A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

We consider non-overlapping subgraphs of fixed order in the random graph K_n, p(n). Fix a strictly strongly balanced graph G. A subgraph of K_{n, p(n)} isomorphic to G is called a G-subgraph. Let X_n be the number of G-subgraphs of K_{n, p(n)} vertex disjoint to all other G-subgraphs. We show that if E[X_n]→∞ as n→, then X_n/E[X_n] converges to 1 in probability. Also, if E[X_n]→c as n→∞, then X_n satisfies a Poisson limit theorem. the Poisson limit theorem is shown using a correlation inequality similar to those appeared in Janson, ?uczak, and Ruciñski[8] and Boppana and Spencer [4].     

MODp‐tests,almost independence and small probability spaces

In this paper, we consider approximations to probability distributions over Z . We present an approach to estimate the quality of approximations to probability distributions towards the construction of small probability spaces. These small spaces are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan, and Veličković [EGLNV], the methods which are used here are simple, and we get smaller sample spaces. Our investigations are motivated by recent work of Azar, Motwani, and Naor [AMN]. They considered the problem to construct in time respective space polynomial in n a good approximation to the joint probability distribution of the mutually independent random variables X₁, X₂,…,X_n. Each X_i has values in {0, 1} and satisfies X_i=0 with probability q and X_i=1 with probability 1−q where q∈[0, 1] is arbitrary. Our considerations improve on results in [EGLNV] and [AMN] for q=1/p and p a prime. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 293–313, 2000     

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Locally Compact Path Spaces

It is shown that the space X^[0,1], of continuous maps [0,1]X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T₁-space X, it follows that X^[0,1] is locally compact if and only if X is locally compact and totally path-disconnected. Mathematics Subject Classifications (2000) 54C35, 54E45, 55P35, 18B30, 18D15.     

Local rigidity of quasi-regular varieties

For a G-variety X with an open orbit, we define its boundary ∂ X as the complement of the open orbit. The action sheaf S _X is the subsheaf of the tangent sheaf made of vector fields tangent to ∂ X. We prove, for a large family of smooth spherical varieties, the vanishing of the cohomology groups H ⁱ (X, S _X ) for i > 0, extending results of Bien and Brion (Compos. Math. 104:1–26, 1996). We apply these results to study the local rigidity of the smooth projective varieties with Picard number one classified in Pasquier (Math. Ann., in press).     

Maximal transitive sets with singularities for genericC 1 vector fields

A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC ¹ generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC ¹ vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq.     

Relatively free bands

The operators h_n and i _n and their duals hmacr;_n and imacr;_n defined on the free semigroup X ⁺ for a nonempty set X by Gerhard and Petrich are proved here to be homomorphisms of onto certain subsets of X ⁺ with a suitable multiplication. From the work of the authors mentioned, these operators induce fully invariant congruences on X ⁺ corresponding to join irreducible varieties of bands if X is countably infinite. New operators on X ⁺ are defined by means of these operators which give homomorphisms in an analogous way and induce fully invariant congruences on X ⁺corresponding to all varieties of bands except for the variety of all bands, and some varieties of normal bands. The former of these was investigated by the mentioned authors and the latter must be treated differently. By means of the above operators we are able to characterize all cases of relatively free bands.     

The Grothendieck ring of varieties and piecewise isomorphisms

Let K ₀(Var _k ) be the Grothendieck ring of algebraic varieties over a field k. Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X ₁, ..., X _n , Y ₁, ..., Y _n into locally closed subvarieties such that X _i is isomorphic to Y _i for all i ≤ n), then [X] = [Y] in K ₀(Var _k ). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1 or X is a smooth connected projective surface or if X contains only finitely many rational curves.     

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study the modificationAA of an affine domainA which produces another affine domainA=A[I/f] whereI is a nontrivial ideal ofA andf is a nonzero element ofI. First appeared in passing in the basic paper of O. Zariski [Zar], it was further considered by E. D. Davis [Da]. In [Ka1] its geometric counterpart was applied to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions (more general than those in [Ka1]) which guarantee preservation of the topology under a modification.As an application, we show that the group of biregular automorphisms of the affine hypersurfaceXC ^k+2, given by the equationuv=(p(x ₁,...,x_k) wherepC[x ₁,...,x _k ],k2, actsm-transitively on the smooth part regX ofX for anymN. We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Partially supported by the NSA grant MDA904-96-01-0012.     

Characterizations of multivariate normality II. Through linear regressions

It is established that a vector (X′₁, X′₂, …, X′_k) has a multivariate normal distribution if (i) for each X_i the regression on the rest is linear, (ii) the conditional distribution of X₁ about the regression does not depend on the rest of the variables, and (iii) the conditional distribution of X₂ about the regression does not depend on the rest of the variables, provided that the regression coefficients satisfy some more conditions that those given by [4]J. Multivar. Anal. 6 81–94].